In the novel Bitterblue, by Kristin Cashore, one character has a watch which is divided into 15 hours, each of which has 50 minutes. The novel has a brief discussion of how to convert time on that watch to standard time, and I'd like to look at it in a little more detail.
Like a conventional watch, which shows 12 hours of 60 minutes, the watch in Bitterblue shows half a day. However, the number of periods, and therefore the lengths of the periods, that it shows are different. One day has 24 standard hours, but 30 watch hours. (I will refer to times and durations on the watch as w-hours, w-minutes, etc. for clarity.) Therefore, there are 4 hours in 5 w-hours, or 1 w-hour = 4/5 hour (or 48 minutes). With 60 minutes in an hour, there are 1440 minutes in a day, but since there are only 50 w-minutes in a w-hour, there are 1500 w-minutes in a day. This means that 24 minutes = 25 w-minutes, so minutes and w-minutes have a similar duration.
With these relationships, it's possible to convert a time on the watch to a conventional time. One way to do it is to convert w-hours to w-minutes, then convert the total w-minutes to conventional minutes, then convert the minutes back to hours and minutes. If the w-time is h:m, the formula to convert to minutes is (h×50 + m) × 24/25. Divide this number by 60 to get the current hour, and the remainder is the current minutes. In Bitterblue, the title character does an example of a similar computation with equivalent results.
After Bitterblue does the calculation, she remarks that "I, for one, would find it simpler to memorize which time signifies what." As a halfway step to memorizing lots of times, it's fairly easy to estimate the time from the w-time. We'll start with a rough estimate that's accurate to about 5 minutes and then tighten it up a bit. 4 hours = 5 w-hours, so 4:00 = 5:00 w-time, 8:00 = 10:00 w-time, and 12:00 = 15:00 w-time. The first step is to find the closest current hour to one of these three points. The second step is to observe that 1 w-hour is 4/5 hour, and that 3/4 is close to 4/5. We're doing some rounding here, but we're used to thinking in quarter hours and we can correct the rounding later if we need to. So we start at 5:00, 10:00, or 15:00, and we add or subtract enough w-hours to be close to the current w-time. For each w-hour added or subtracted, we add or subtract 3/4 hour from the time. The last step is to add or subtract w-minutes. Since we're just estimating, and 1 w-minute = 24/25 minute, we can just add or subtract the number of w-minutes after or before the hour and ignore the conversion.
Let's do an example. Say the w-time is 8:35. 8 is close to 10, so we start there. 10:00 w-time = 8:00. Then we subtract 1 w-hour from 10:00 to get 9:00, so we subtract 45 minutes from 8:00 to get 7:15. Finally, we subtract 15 w-minutes from 9:00 to get 8:35. (The minutes subtraction is the step which throws me. Since there are 50 w-minutes in a w-hour, 8:35 is 15 w-minutes before the hour, not the expected 25.) Subtracting 15 minutes from 7:15 gives our estimate of 7:00. 8:35 on the watch is approximately 7:00 normal time.
We did some rounding, which we can now correct if we need more precision. We approximated 1 w-hour as 3/4 hour, when it's really 4/5 hour. 3/4 is 45 minutes and 4/5 hour is 48 minutes, so we can add or subtract 3 additional minutes per w-hour. In this case, 6:57 is a closer estimate than 7:00. Finally, there's a small rounding error in the minutes. If we add close to 25 w-minutes, we should subtract 1 minute from our estimate, and vice versa. Since we subtracted 15 w-minutes, it's slightly closer to add 1 minute back in, for a final time of 6:58.
Doing the full computation, we get 8×50 + 35 = 435. 435×24 is 10440. 10440/25 is 417.6. 417.6 divided by 60 is 6, with a remainder of 57.6. In other words, the exact time is 6:57:36. I can do 435×24 in my head, but I don't really want to, and estimating that the current time is about 7:00 was much easier and probably accurate enough.
Often, when I look at a watch face, I don't actually need to know the exact time. I'm just looking for a quick estimate, based on the hand position. So what does the hand position on a 15 hour watch tell us about the standard time? On a standard watch, the hour hand travels a full circle in one half day, so the angle from vertical tells us the exact time. It's easy to judge that if the hour hand is pointing down and a little to the left, it's about 7:00, and based on the exact angle it's not hard to judge whether it's closer to 6:30, 7:00, or 7:30, without even referring to the minute hand.
On the 15 hour watch, the hour hand travels a full circle in one half day, exactly the same as a standard watch. So the angle of the hour hand is the same as on the standard watch. You can work out the angles from the example, and you will find that the angle of the hour hand at 8:35 on the 15 hour watch is nearly the same as the angle of the hour hand at 7:00 on a standard watch. This means that a quick glance at the hour hand on a 15 hour watch will give you exactly the same information as a quick glance at the hour hand on a standard watch.
The minute hand is a different story. The minute hand on a 15 hour watch completes one full circle in 48 (standard) minutes, which means that it points all kinds of different directions relative to the standard minute hand. In our example, at 8:35, the minute hand is pointing to the left and slightly down on the 15 hour watch, but at 7:00 on a standard watch it's pointing straight up. It's hard to get useful information out of the minute hand without doing the full time conversion, either estimated or using the exact formula.
One other point about the watch face design on the 15 hour watch: On a standard watch, it's possible for one set of marks to indicate both hours and minutes. The angle representing 1 hour on the hour hand is the same as the angle representing 5 minutes on the minute hand. So you can mark just the hours, and it's easy to read the minutes just off the hour markings. On the 15 hour watch, things don't work out so well. The angle representing 1 w-hour is the angle for 3 1/3 w-minutes. To read this watch with the same ease and precision as a standard watch, you really need two sets of marks, one for w-hours and one for w-minutes.
Friday Random Ten
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #183, "Sie Werden Euch In Den Bann Tun"
- NDR Chorus Hamburg - Johannes Brahms: 12 Lieder Und Romanzen, Op. 44, "Fragen"
- Johannes Brahms: Neue Liebeslieder Waltzer, Op. 65, "Nein, Geliebter, Setze Dich"
- Wiener Philharmoniker - Ludwig van Beethoven: Fidelio, Op. 72, Overture
- Moby - The Sky Is Broken
- Birdsongs Of The Mesozoic - Ptinct
- Mozart Akademie Amsterdam - Wolfgang Amadeus Mozart: Symphony #36 "Linz", Presto
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #27, "Willkommen! Will Ich Sagen"
- Tori Amos - Cruel
- Frank Martin: Mass for Double Choir, Credo
Sequences have come up previously in the discussion of topology, and they may have also come up back in February when I was discussing real numbers, but I'm not sure if I've ever formally defined sequences. It turns out that the definition is really simple. A sequence is a function which has a domain of the natural numbers. We could write the sequence as f(n), using n to remind us that it's just defined for natural numbers, but we usually write it as an instead.
The codomain of the sequence could potentially be anything. In the development of topology, the codomain of the sequence was often a set in some metric space. In the proof that perfect sets in Euclidean spaces are uncountable, we used two sequences of points in the Euclidean space, a sequence of neighborhoods, and a sequence of closed sets. Now that our focus is on sequences, the codomain will always be individual points in a metric space, and often in a more specific space such as the complex plane or the real number line.
The big question with a sequence is where is it going? As n gets big, does the sequence of points have some tendency? There are three basic possibilities, depending on both the sequence and the space. First, the sequence could become near a single point. Second, for a sequence of real numbers, the sequence could just increase forever. Third, the sequence could never settle down in a single direction.
If the sequence gets and stays near a single point, we can say that the sequence converges to that point. Convergence has a strict definition, which I think is beautiful, even though some people seem to hate it. If we are in some metric space, which has some distance function d(p,q) between any two points in the metric space, any convergent sequence converges to a point L, which is called the limit of the sequence. Just because L is the limit of a sequence doesn't mean that any point in the sequence actually equals L. It just means that the points in the sequence get close to L, and that they get closer as n gets bigger.
For example, consider the sequence defined by an=1/n. We are of course familiar with this as a set from topology, but the set doesn't necessarily have a particular ordering. We are now looking at it as a sequence, which means that it is a function with domain of the natural numbers, and has an order based on the order of the natural numbers. It should be clear, both from the discussion of topology and from general observation, that the limit of this sequence is 0. That is, as n increases, an gets closer to 0. No point in the sequence ever equals 0, but that's okay because the sequence gets as close as you would like to 0.
I keep saying the sequence "gets near" the limit. It's fair to ask, how near? The answer is, as near as you want to get. Pick some distance from the limit, and call that distance ε. Then we are interested in values of n such that d(an,L) is less than ε. In particular, we are interested in a minimum value N, such that for all natural numbers greater than N, d(an,L)<ε. Looking at the sequence an=1/n, if we picked ε=1/100, for example, then d(an,0)<ε for any value of n≥101. If you choose a different ε, there's a different start point, but for any positive distance ε, there is always a minimum value of N, such that for any n≥N, the distance from an to L is always less than ε.
And this is the formal definition of the limit of sequence. A sequence an in some metric space has limit L, if for any positive real number ε, there exists a natural number N, such that for any value of n greater or equal to N, the distance from an to L is less than ε. Symbolically, the last sentence is equivalent to (∀ε>0)(∃N∈ℕ)(∀n≥N) d(an,L)<ε.
A straightforward conclusion from the definition of a limit is that if a sequence has a limit, it must be unique. This can be proved by assuming that the sequence has two limits, and then using ε and the triangle theorem. If a sequence an has limits L and M, then d(L,M)≤d(an,L)+d(an,M)<2ε. Since ε is arbitrarily small, the distance between L and M is arbitrarily small, which means they must be equal, because d(L,M)=0 only if L=M. You can also argue that if L is the limit of the sequence, no other point can be the limit, because the sequence always gets within distance ε of L, and since any other point is a fixed distance from L, you can always choose ε less than the distance to that point.
Now that I've developed enough topology to prove that the real numbers are not countable, I'm ready to move on to the next topic, which is sequences and series. Before I get into the details, I want to talk about some general issues around the subject.
First off, the book specifically develops topology before sequences because it requires topology concepts in order to provide the structure for discussing sequences. On the other hand, I ended up using sequences frequently in my discussion of topology. I didn't rely on a lot of theory of sequences, but I did use an implicit definition of sequences. If you read some of my proofs revolving around sequences of sets and were wondering how exactly sequences were defined, I will clarify that now. At the same time, there were certain proofs where I desperately wanted to use more theory of sequences, and I avoided doing so because I hadn't developed the theory yet. I don't think greater use of sequences would have made my posts about topology more rigorous, but it may have made it easier to understand.
I've previously mentioned the idea that studying any particular subject in math leads inevitably to studying every subject in math. The way that sequences and topology are tangled up with each other leads to the conclusion that not only do you have to study every subject, but you have to study every subject at the same time. They all build each other. As another example, sequences and series are usually introduced as topics in Calculus 2, when you can use concepts based on integration to prove the basic theorems. On the other hand, the proof that integration works is dependent on Riemann sums, which are infinite series. This sort of thing could turn into circular reasoning, but it can also turn into both topics providing scaffolding for the other, where the results are rigorous but dependent on developing both subjects simulateously.
Speaking of Calculus 2, when studying sequences and series in Calculus 2, I felt like the material both didn't do enough and was looking to do too much. The important question is whether or not a particular sequence or series converges, not what that sequence or series converges to. This was frustrating to me, because it feels like we're only solving half the problem. At the same time, the tools for determining convergence feel weak, making applying them difficult. It's like we're choosing to only solve half the problem, and then handicapping ourselves to make the problem hard.
The approach we are now taking to sequences and series doesn't require calculus, which is good. But it doesn't really prove any more than calculus did, which is bad. We're still just asking whether a sequence converges, and we still end up with a similar set of tools. At least the results are consistent with Calculus 2, so there's at least that.
As I've been studying analysis, I've become convinced that sequences are the basis for everything that follows. Even when the presentation feels incomplete or weak, it's important to understand sequences in order to support analysis as a whole. So this will be my next major topic on this blog.